Goal: Identify the joint distribution of the latent failure times
given that we only observe the distribution of the
identified minimum.

Note that we aren’t considering regressors yet.

Cox and Tsiatis Nonidentification Theorem

For any joint distribution of latent failure times, there exists
another such distribution with independent failure times that yields
the same distribution of the minimum (Cox, 1959, 1962; Tsiatis, 1975).

That is, given r.v.’s (T1,T2,…,TJ) there exist (S1,S2,…,SJ) with Si⫫Sj for all i≠j such that (T,IT)
and (S,IS) are observationally equivalent.

In light of this result, any empirical work needed to proceed by
placing some structure on the form of dependence across risks, for example,
by assuming independence.

Importance of Dependence

We are concerned with conditional independence—independence of the
risks T1,…,TJ conditional on X.

Even conditional independence may not hold if, for example, we are
studying an individual whose behavior may affect all of the risks.

Yashin, Manton, and Stallard (1986): How do smoking, blood pressure,
and body weight (regressors) affect time of death from cancer, heart
disease, etc. (risks).

Overview

Establish an identification theorem for a general class of competing
risks models with regressors.

This class includes models with marginal distributions that follow:

Proportional hazards.

Mixed proportional hazards.

Accelerated hazards.

Results are presented for only two competing risks but generalize to
any arbitrary finite number of risks.

Proportional Hazards Model

We want to model the time of death T from a single risk
conditional on some covariates X.

Conditional on X, T has cdf F(t|x) and pdf f(t|x).

Hazard function: λ(t|x)=f(t|x)1−F(t|x).

Integrated hazard: Λ(t|x)=∫0tλ(s|x)ds.

If λ(t|x)=z(t)ϕ(x) then Λ(t|x)=Z(t)ϕ(x) with
Z(t)=∫0tz(s)ds.

Equivalently, we can work with the survivor function:
S(t|x)=Pr(T>t|x)=exp[−Z(t)ϕ(x)]

It is common in practice to use ϕ(x)=exβ.

Suppose F(t|x)=1−e−Z(t)ϕ(x) where Z(t) is the
baseline integrated hazard and ϕ(x) is a scaling term.

If Z is differentiable, then Z′(t) is the baseline hazard.

Proportional Hazards and Competing Risks

Assuming for the moment that failure times are independent,
we can easily generalize this to model competing risks.

The distribution of each failure time has a proportional hazard
specification.

Z(t) and ϕ may differ across risks.

The joint survivor function is
S(t1,t2|x)=1−(1−exp[−Z1(t1)ϕ1(x)])(1−exp[−Z2(t2)ϕ2(x)]).

Introducing Dependence

We could draw two independent failure times T1 and T2 by
drawing (independently) Uj∼U(0,1) and solving for Tj:

Sj(t|x)=exp{−Zj(t)ϕj(x)}.

If K(u1,u2)=u1u2 is the CDF of U1 and U2, the joint
survivor function is

S(t1,t2|x)=K[exp{−Z1(t)ϕ1(x)},exp{−Z2(t)ϕ2(x)}].

We can introduce dependence in T1 and T2 by introducing
dependence in U1 and U2 via K.

Suppose (U1,U2)∼K(⋅,⋅) on [0,1]2 and
assume that Z1(0)=Z2(0)=0.

Then the survivor function for (T1,T2) is

(1)S(t1,t2|x)=K(exp[−Z1(t1)ϕ1(x)],exp[−Z2(t2)ϕ2(x)]).

Generalization: Mixed Proportional Hazards

Suppose that the competing risks are independent, ϕj(x)=exβj,
and that one of the covariates, ω, is not observed:

S(t1,t2|x)=∫Ωexp[−Z1(t1)exβ1+c1ω]exp[−Z2(t2)exβ2+c2ω]dG(ω).

We can arrive at this model by choosing K such that:

K(η1,η2)=∫Ωη1exp(c1ω)η2exp(c2ω)dG(ω).

Generalization: Accelerated hazards

S(t|x)=exp[−Z{tϕ(x)}]

Joint survivor with dependent competing risks:

S(t1,t2|x)=K(exp[−Z1{t1ϕ1(x)}],exp[−Z2{t2ϕ2(x)}]).

For any K, the marginal distributions give rise to univariate
accelerated hazard models.

Identification Theorem

Assume that (T1,T2) has joint distribution (1).
Then Z1, Z2, ϕ1, ϕ2, and K are identified from
the minimum of (T1,T2) under the following assumptions:

K is continuously differentiable with partial derivatives K1
and K2 and for i=1,2, limn→∞Ki(η1n,η2n)
is finite for all sequences η1n, η2n for which
η1n→1 and η2n→1 for n→∞. We also
assume that K is strictly increasing in each of its arguments.

Z1(1)=Z2(1)=1 and ϕ1(x0)=ϕ2(x0)=1 for some
x0.

The support of {ϕ1(x),ϕ2(x)} is
(0,∞)×(0,∞).

Z1 and Z2 are nonnegative, differentiable, strictly increasing
functions, except that we allow them to be infinite for finite t.

Notes about these assumptions:

K is already weakly increasing.

This is an innocuous normalization since ϕj(x) and Zj(t) are
not jointly identified to scale.

This is satisfied, for example, when ϕj(x)=exp(xβj)
and there is a common covariate with support ℝ and different
coefficients.

Mapping Observables to Unobservables

Observed distributions:

Q1(t|x)=Pr(T1≥t,T2≥T1|x)Q2(t|x)=Pr(T2≥t,T1≥T2|x).

Tsiatis (1975) establishes the following mappings:

∂Q1∂t(t|x)=[∂S∂t1]t1=t2=t∂Q2∂t(t|x)=[∂S∂t2]t1=t2=t.

We have
∂Q1∂t(t|x)=−K1[exp{−Z1(t)ϕ1(x)},exp{−Z2(t)ϕ2(x)}]exp{−Z1(t)ϕ1(x)}Z′1(t)ϕ1(x).